An invariant-region-preserving scheme for a convection-reaction-Cahn-Hilliard multiphase model of biofilm growth in slow sand filters

TL;DR

The paper addresses modeling biofilm growth in slow sand filters by coupling a convective, degenerate-mobility Cahn–Hilliard equation to a volume-averaged Stokes flow for a multiphase mixture.It develops an invariant-region-preserving upwind discontinuous Galerkin scheme with a reaction-splitting time discretization to guarantee nonnegativity and an upper bound on the total solid concentration.The model is extended from a one-dimensional to a two-dimensional setting, introducing multiple solid and liquid subphases and deriving the coupled CH–Stokes system with a mass-conserving velocity field solver.Numerical experiments validate positivity, mass conservation, and boundedness, and illustrate biofilm growth in the supernatant water under various reaction regimes, including inflow/outflow boundary scenarios.

Abstract

A multidimensional model of biofilm growth present in the supernatant water of a Slow Sand Filter is derived. The multiphase model, consisting of solid and liquid phases, is written as a convection-reaction system with a Cahn-Hilliard-type equation with degenerate mobility coupled to a Stokes-flow equation for the mixture velocity. An upwind discontinuous Galerkin approach is used to approximate the convection-reaction equations, whereas an $H^1$-conforming primal formulation is proposed for the Stokes system. By means of a splitting procedure due to the reaction terms, an invariant-region principle is shown for the concentration unknowns, namely non-negativity for all phases and an upper bound for the total concentration of the solid phases. Numerical examples with reduced biofilm reactions are presented to illustrate the performance of the model and numerical scheme.

Figures (5)

• Figure 4.1: Interaction of two circular masses a distance apart without any reactions: Biofilm concentration $\widetilde{u}$ and mixture velocity $\boldsymbol{q}$ (arrows) with zero boundary conditions at six time points, with $R_1=R_2=0$. Both initial circles have the same radius $0.2$, and are initially centred at $(0.35, 0.75)^\texttt{t}$ and $(0.65, 0.35)^\texttt{t}$, respectively.
• Figure 4.2: Interaction of two circular masses closer together: Biofilm concentration $\widetilde{u}$ and mixture velocity $\boldsymbol{q}$ (arrows) with zero boundary conditions for the case without reactions (top) and with reactions $R_1 = 100$ and $R_2 = 1000$ (bottom) at three time points. Both the initial discs of masses have the same radius $0.2$, and are centred at $(0.45, 0.75)^\texttt{t}$ and $(0.55, 0.35)^\texttt{t}$, respectively.
• Figure 4.3: Interaction of two circular masses: Comparison of the approximated $\widetilde{u}$ with $\gamma=0$ (left plot) and $\gamma=1$ (middle plot) at $t= 8\times10^{-3}\,\rm s$. Right plot: Schematic of the supernatant water region of an SSF with inflow at the top boundary in $\Gamma_{\rm in}$ and outflow at the bottom boundary $\Gamma_{\rm out}$. On $\Gamma_{\rm wall}$, no-slip boundary conditions are considered.
• Figure 4.4: Simulations of an SSF: Total biofilm concentration $\widetilde{u}$ and mixture velocity $\boldsymbol{q}$ (arrows) with in- and outflow boundary conditions for zero, medium and large reactions (columns). The time points (rows) are $t_3 \coloneqq 3\times10^{-3}\,\rm s$, $t_6 \coloneqq 6\times10^{-3}\,\rm s$ and $t_9 \coloneqq 9\times10^{-3}\,\rm s$.
• Figure 4.5: Simulation of an SSF: Component concentrations $c^{(1)}$ (1st column), $c^{(2)}$ (2nd column), $s^{(1)}$ (3rd column) and $s^{(2)}$ (4th column) for the case of in- and outflow boundary conditions and $5R_1=R_2 = \qty{100}{\per\ourtime}$. The time points are $t_1 = 10^{-3}\,\rm s$, $t_3 = 3\times10^{-3}\,\rm s$, $t_6 = 6\times10^{-3}\,\rm s$ and $t_9 = 9\times10^{-3}\,\rm s$. The black contour line corresponds to the solid-liquid interface of $\widetilde{u}$.

Theorems & Definitions (22)

• Lemma 3.1
• proof
• Theorem 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4
• proof
• Lemma 3.5
• proof